Affine Springer fibers of type A and combinatorics of diagonal coinvariants

Abstract We calculate the Borel–Moore homology of affine Springer fibers of type A associated with some regular semisimple nil elliptic elements. As a result, we obtain bigraded S n -modules whose bigraded Frobenius series are a generalization of the symmetric functions introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov.

[1]  E. Sommers.  A family of affine Weyl group representations , 1997 .

[2]  T. Shoji.  Geometry of orbits and Springer correspondence , 1988 .

[3]  I. Gessel.  Multipartite P-partitions and inner products of skew Schur functions , 1983 .

[4]  B. Sagan.  The Symmetric Group , 2001 .

[5]  N. Chriss,et al.  Representation theory and complex geometry , 1997 .

[6]  D. Kazhdan,et al.  Fixed point varieties on affine flag manifolds , 1988 .

[7]  Alain Lascoux,et al.  Ribbon tableaux, Hall–Littlewood functions, quantum affine algebras, and unipotent varieties , 1995 .

[8]  George Lusztig,et al.  Fixed Point Varieties on the Space of Lattices , 1991 .

[9]  Shrawan Kumar,et al.  Kac-Moody Groups, their Flag Varieties and Representation Theory , 2002 .

[10]  A. Moy,et al.  Unrefined minimal K-types forp-adic groups , 1994 .

[11]  Fibres de Springer et jacobiennes compactifiées , 2002, math/0204109.

[12]  Jens Piontkowski,et al.  Topology of the compactified Jacobians of singular curves , 2006 .

[13]  Corrado De Concini,et al.  Symmetric functions, conjugacy classes and the flag variety , 1981 .

[14]  Mark D. Haiman.  ON THE QUOTIENT RING BY DIAGONAL INVARIANTS , 1994 .

[15]  Jean-Yves Thibon,et al.  Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials , 1998, math/9809122.

[16]  Mark Haiman,et al.  Vanishing theorems and character formulas for the Hilbert scheme of points in the plane , 2001, math/0201148.

[17]  Mark Haiman,et al.  Combinatorics, symmetric functions, and Hilbert schemes , 2002 .

[18]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[19]  T. Tanisaki.  Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups , 1982 .

[20]  Evgeny Gorsky,et al.  Compactified Jacobians and q,t-Catalan numbers, II , 2012, Journal of Algebraic Combinatorics.

[21]  G. Lusztig.  Affine weyl groups and conjugacy classes in Weyl groups , 1996 .

[22]  J. B. Remmel,et al.  A combinatorial formula for the character of the diagonal coinvariants , 2003, math/0310424.